The core in Participatory Budgeting can be empty
证明在基于审批的参与式预算中，对于大多数基于项目成本的满意度函数，核心可能为空，但基数满意度函数的情况仍悬而未决。
The core is a strong fairness notion in multi-winner voting and participatory budgeting (PB). It is known that the core can be empty if we consider cardinal utilities, but it is not known whether it is always satisfiable with approval ballots. In this short note, I show that in approval-based PB the core can be empty for nearly all satisfaction functions that are based on the cost of a project. In particular, I show that the core can be empty for the cost satisfaction function, satisfaction functions based on diminishing marginal returns, and the share. However, it remains open whether the core can be empty for the cardinality satisfaction function. • The core in participatory budgeting can be empty. • The presented counter-example covers most relevant satisfaction functions. • It remains open whether the result holds for the cardinal satisfaction function.